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Properties

Label 2-21e2-63.20-c1-0-19
Degree $2$
Conductor $441$
Sign $-0.183 - 0.983i$
Analytic cond. $3.52140$
Root an. cond. $1.87654$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 + 0.934i)2-s + (1.01 + 1.40i)3-s + (0.746 + 1.29i)4-s + (1.25 + 2.17i)5-s + (0.326 + 3.22i)6-s − 0.947i·8-s + (−0.948 + 2.84i)9-s + 4.68i·10-s + (−4.85 − 2.80i)11-s + (−1.06 + 2.35i)12-s + (0.384 − 0.221i)13-s + (−1.78 + 3.95i)15-s + (2.37 − 4.11i)16-s + 3.07·17-s + (−4.19 + 3.71i)18-s − 2.57i·19-s + ⋯
L(s)  = 1  + (1.14 + 0.660i)2-s + (0.584 + 0.811i)3-s + (0.373 + 0.646i)4-s + (0.560 + 0.970i)5-s + (0.133 + 1.31i)6-s − 0.335i·8-s + (−0.316 + 0.948i)9-s + 1.48i·10-s + (−1.46 − 0.845i)11-s + (−0.306 + 0.680i)12-s + (0.106 − 0.0615i)13-s + (−0.459 + 1.02i)15-s + (0.594 − 1.02i)16-s + 0.746·17-s + (−0.988 + 0.876i)18-s − 0.590i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.183 - 0.983i$
Analytic conductor: \(3.52140\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :1/2),\ -0.183 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90875 + 2.29786i\)
\(L(\frac12)\) \(\approx\) \(1.90875 + 2.29786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.01 - 1.40i)T \)
7 \( 1 \)
good2 \( 1 + (-1.61 - 0.934i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.25 - 2.17i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.85 + 2.80i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.384 + 0.221i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.07T + 17T^{2} \)
19 \( 1 + 2.57iT - 19T^{2} \)
23 \( 1 + (-6.83 + 3.94i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.71 - 1.56i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9.06 - 5.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.41T + 37T^{2} \)
41 \( 1 + (-1.64 - 2.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.75 - 8.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.07 - 1.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.85iT - 53T^{2} \)
59 \( 1 + (3.65 + 6.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.40 - 4.27i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.934 - 1.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.95iT - 71T^{2} \)
73 \( 1 + 8.51iT - 73T^{2} \)
79 \( 1 + (-0.287 + 0.497i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.23 + 7.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 7.57T + 89T^{2} \)
97 \( 1 + (3.22 + 1.86i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.06613733442946008706961056561, −10.56469121293740128347836682843, −9.716503885077937497361419532968, −8.555098745817901062247683683878, −7.49064530496851616724351493809, −6.51187479126122336880126540749, −5.43233525276322781485142369342, −4.85377319872067703843523492419, −3.30996121712034905831727467017, −2.83263734590900804589183186349, 1.58294486401363621157180568942, 2.60203313878113145965116333061, 3.78579834501432874044113145038, 5.18484413109573456432590176014, 5.60692760437110114216620286364, 7.22468699290861547810387219630, 8.090377932651907301531361028289, 9.051718095632482912575720659467, 10.03364017437486997820725591091, 11.17912438532181157199979032543

Graph of the $Z$-function along the critical line